On the continuity of non constant functions

Question

Suppose $f$ is a non constant continuous function. Doesn't $f(x+\epsilon) \neq f(x)$ for some $\epsilon \to 0$ imply that we need to "lift our pen to draw the next point" and thus $f$ can't be continuous? I mean it in the sens that, were we to be able to have a just next point to $x$, there will be a "jump". This is of course necessary for a function to be non constant, but suppose $f(x)=x^\alpha$ with $\alpha \to \infty$, that jump would be considerable.